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A review on the products of distributions

  • C. K. Li

Abstract

The problem of defining products of distributions has been open and an active research area since Schwartz introduced the theory of distribution around 1950. The inherent difficulties of obtaining products have never prevented their appearance in literature, as they are needed in quantum field and differential equations with distribution involved. The objective of this paper is to recollect various approaches, which include sequential and complex analysis methods, to tackling products of distributions in one or multiple variables, as well as particular generalized functions defined on certain manifolds.

Keywords

Integral Transform Laurent Series Summable Function Laurent Expansion Elementary Particle Physic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Sch59]
    Schwartz, L.: Théorie des distributions. Vols.I, II, Hermann, Paris (1957)MATHGoogle Scholar
  2. [Gasi66]
    Gasiorowicz, S.: Elementary particle physics. J. Wiley and Sons Inc., New York (1966)Google Scholar
  3. [AMS73]
    Antosik, P., Mikusinski, J. and Sikorski, R.: Theory of distributions, the sequential approach. PWN-Polish Scientific Publishers, Warsawa (1973)MATHGoogle Scholar
  4. [Bre65]
    Bremermann, J.H.: Distributions, complex variables, and Fourier transforms. Addison-Wesley, Reading, Massachusetts (1965)MATHGoogle Scholar
  5. [Li78]
    Li, B.H.: Non-standard analysis and multiplication of distributions. Sci. Sinica, 21(5), 561–585 (1978)MathSciNetGoogle Scholar
  6. [EGO92]
    Embacher, H.G., Grübl, G., Oberguggenberger, M.: Z. Anal. Anw., 11, 437–454 (1992)MATHGoogle Scholar
  7. [GS64]
    Gel’fand, I.M., Shilov, G.E.: Generalized functions. Vol. I. Academic Press, New York London (1964)MATHGoogle Scholar
  8. [Fis71]
    Fisher, B.: The product of distributions. Quart. J. Math. Oxford, 22, 291–298 (1971)MATHCrossRefGoogle Scholar
  9. [Fis74]
    Fisher, B.: The neutrix distribution product x +r δ (r−1). Studia Sci. Math. Hungar., 9, 439–441 (1974)MathSciNetGoogle Scholar
  10. [Fis82a]
    Fisher, B.: On defining the convoltion of distributions. Math. Nachr., 106, 261–269 (1982)MATHCrossRefMathSciNetGoogle Scholar
  11. [Fis82b]
    Fisher, B.: A non-commutative neutrix product of distributions. Math. Nachr., 108, 117–127 (1982)MATHCrossRefMathSciNetGoogle Scholar
  12. [Fis80]
    Fisher, B.: On defining the product of distributions. Math. Nachr., 99, 239–249 (1980)MATHCrossRefMathSciNetGoogle Scholar
  13. [KF03]
    Kilicman, A., Fisher, B.: On the Fresnel integrals and the convolution. Int. J. Math. Math. Sci., 41, 2635–2643 (2003)CrossRefMathSciNetGoogle Scholar
  14. [FT05a]
    Fisher, B., Taş, K.: The convolution of functions and distributions. J. Math. Anal. Appl., 306(1), 364–274 (2005)MATHCrossRefMathSciNetGoogle Scholar
  15. [FT06a]
    Fisher, B., Taş, K.: On the composition of the distributions x +λ and x +μ. J. Math. Anal. Appl., 318(1), 102–111 (2006)MATHCrossRefMathSciNetGoogle Scholar
  16. [FT05b]
    Fisher, B., Taş, K.: On the non-commutative neutrix product of the distributions x r lnp |x| and x −s. Integral Transform Spec. Funct., 16(2), 131–138 (2005)MATHCrossRefGoogle Scholar
  17. [FT06b]
    Fisher, B., Taş, K.: On the commutative product of distributions. J. Korean Math. Soc., 43(2), 271–281 (2006)MATHMathSciNetGoogle Scholar
  18. [FN98]
    Fisher, B., Nicholas, J.D.: Some results on the commutative neutrix product of distributions. J. Anal., 6, 33–44 (1998)MATHMathSciNetGoogle Scholar
  19. [Fis72]
    Fisher, B.: The product of the distributions x +r−1/2 and x r−1/2. Proc. Cambridge Philos. Soc., 71, 123–130 (1972)MATHMathSciNetGoogle Scholar
  20. [FL01]
    Fisher, B., Li, C.K.: On the cosine and sine integrals. Int. J. Appl. Math., 7(4), 419–437 (2001)MathSciNetMATHGoogle Scholar
  21. [FL93]
    Fisher, B., Li, C.K.: A commutative neutrix convolution product of distributions. Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 23(1), 13–27 (1993)MATHMathSciNetGoogle Scholar
  22. [FKL00]
    Fisher, B., Kilicman, A., Li, C.K.: An extension of a result on the non-commutative neutrix convolution product of distributions. Int. J. Appl. Math., 3(1), 71–80 (2000)MATHMathSciNetGoogle Scholar
  23. [Fis70]
    Fisher, B.: The generalized function (x + i0)λ. Proc. Camb. Phil. Soc., 68, 707–708 (1970)MATHGoogle Scholar
  24. [FÖG05]
    Fisher, B., Özçag, E., Gülen, Ü.: A theorem on the commutative neutrix product of distributions. Sarajevo J. Math., 1, 235–242 (2005)MathSciNetGoogle Scholar
  25. [vdC60]
    van der Corput, J.G.: Introduction to the neutrix calculus. J. Analyse Math., 7, 291–398 (1959)CrossRefGoogle Scholar
  26. [Li00]
    Li, C.K.: The product of r −k and ∇δ. Int. J. Math. Math. Sci., 24, 361–369 (2000)MATHCrossRefMathSciNetGoogle Scholar
  27. [Li01a]
    Li, C.K.: A note on the product r k · ∇(Δr 2−m). Integral Transform. Spec. Func., 12, 341–348 (2001)MATHCrossRefGoogle Scholar
  28. [LF:90]
    Li, C.K., Fisher, B.: Examples of the neutrix product of distributions on R m. Rad. Mat., 6, 129–137 (1990)MATHMathSciNetGoogle Scholar
  29. [CL05a]
    Li, C.K.: The products on the unit sphere and even-dimension spaces. J. Math. Anal. Appl., 305(1), 97–106 (2005)MATHCrossRefMathSciNetGoogle Scholar
  30. [Li05b]
    Li, C.K.: An approach for distributional products on R m. Integral Transforms Spec. Funct., 16(2), 139–151 (2005)MATHCrossRefMathSciNetGoogle Scholar
  31. [CL91]
    Cheng, L.Z. and Li, C.K.: A commutative neutrix product of distributions on R m. Math. Nachr., 151, 345–355 (1991)MATHCrossRefMathSciNetGoogle Scholar
  32. [CL88]
    Cheng, L.Z. and Li, C.K.: The product of generalized functions. J. Math. Res. Exposition, 8(4), 543–546 (1988)MATHMathSciNetGoogle Scholar
  33. [LK98]
    Li, C.K., Koh, E.L.: The neutrix convolution product in Z’(m) and the exchange formula. Int. J. Math. Math. Sci., 21(4), 695–700 (1998)MATHCrossRefMathSciNetGoogle Scholar
  34. [LZ04]
    Li, C.K., Zou, V.: On defining the product r −k · ∇l δ. Int. J. Math. Math. Sci., 16(13–16), 833–845 (2004)CrossRefMathSciNetGoogle Scholar
  35. [Li01b]
    Li, C.K.: The sequential approach to the product of distribution. Int. J. Math. Math. Sci., 28(12), 743–751 (2001)MATHCrossRefMathSciNetGoogle Scholar
  36. [KL93]
    Koh, E.L., Li, C.K.: On defining the generalized functions δ α(z) and δ n(x). Int. J. Math. Math. Sci., 16(4), 749–754 (1993)MATHCrossRefMathSciNetGoogle Scholar
  37. [Li03]
    Li, C.K.: The neutrix square of δ. Int. J. Appl. Math., 12(2), 115–124 (2003)MATHMathSciNetGoogle Scholar
  38. [LA04]
    Li, C.K., Aguirre, M.A.: The distributional products by the Laurent series. submitted.Google Scholar
  39. [KL92]
    Koh, E.L., Li, C.K.: On the distributions δ k and (δ’)k. Math. Nachr., 157, 243–258 (1992)MATHMathSciNetCrossRefGoogle Scholar
  40. [AL05]
    Aguirre, M.A., Li, C.K.: The distributional products of particular distributions. to appear in Applied Mathematics and Computation.Google Scholar
  41. [Agui03a]
    Aguirre, M.A.: A convolution product of (2j)-th derivative of Diracs delta in r and multiplicative distributional product between r −k and ∇(Δj δ). Int. J. Math. Math. Sci., 13, 789–799 (2003)MathSciNetGoogle Scholar
  42. [Agui03b]
    Aguirre, M.A.: The expansion in series (of Taylor Types) of (k−1) derivatrive of Diracs delta in m 2 + P. Integral Transform. Spec. Func., 14, 117–127 (2003)MATHCrossRefMathSciNetGoogle Scholar
  43. [Agui91]
    Aguirre, M.A.: The series expansion of δ (k)(rc). Mathematicae Notae, 35, 53–61 (1991)MATHMathSciNetGoogle Scholar
  44. [Öz01]
    Özçag, E.: Defining the kth powers of the Dirac-delta distribution for negative integers. Appl. Math. Lett., 14, 419–423 (2001)MATHCrossRefMathSciNetGoogle Scholar
  45. [Ler57]
    Leray, J.: Hyperbolic differential eqautions. The Institute for Advanced Study, Princeton New Jersey (1957)Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • C. K. Li
    • 1
  1. 1.Department of Mathematics and Computer ScienceBrandon UniversityBrandonCanada

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